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Simplifying 4n2 + -7n + -22 = 0 Reorder the terms: -22 + -7n + 4n2 = 0 Solving -22 + -7n + 4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5.5 + -1.75n + n2 = 0 Move the constant term to the right: Add '5.5' to each side of the equation. -5.5 + -1.75n + 5.5 + n2 = 0 + 5.5 Reorder the terms: -5.5 + 5.5 + -1.75n + n2 = 0 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + -1.75n + n2 = 0 + 5.5 -1.75n + n2 = 0 + 5.5 Combine like terms: 0 + 5.5 = 5.5 -1.75n + n2 = 5.5 The n term is -1.75n. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75n + 0.765625 + n2 = 5.5 + 0.765625 Reorder the terms: 0.765625 + -1.75n + n2 = 5.5 + 0.765625 Combine like terms: 5.5 + 0.765625 = 6.265625 0.765625 + -1.75n + n2 = 6.265625 Factor a perfect square on the left side: (n + -0.875)(n + -0.875) = 6.265625 Calculate the square root of the right side: 2.503123049 Break this problem into two subproblems by setting (n + -0.875) equal to 2.503123049 and -2.503123049.Subproblem 1
n + -0.875 = 2.503123049 Simplifying n + -0.875 = 2.503123049 Reorder the terms: -0.875 + n = 2.503123049 Solving -0.875 + n = 2.503123049 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + n = 2.503123049 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + n = 2.503123049 + 0.875 n = 2.503123049 + 0.875 Combine like terms: 2.503123049 + 0.875 = 3.378123049 n = 3.378123049 Simplifying n = 3.378123049Subproblem 2
n + -0.875 = -2.503123049 Simplifying n + -0.875 = -2.503123049 Reorder the terms: -0.875 + n = -2.503123049 Solving -0.875 + n = -2.503123049 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + n = -2.503123049 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + n = -2.503123049 + 0.875 n = -2.503123049 + 0.875 Combine like terms: -2.503123049 + 0.875 = -1.628123049 n = -1.628123049 Simplifying n = -1.628123049Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.378123049, -1.628123049}
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